Classification of indecomposable states on the infinite symmetric inverse semigroup invariant under the infinite symmetric group. Semifinite case

2025/08/19 by Artem Dudko, Dudko, Artem, Nessonov, Nikolay I.
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #semigroups and automata theory #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2508.13760

Abstract

Let ℕ be a set of the natural numbers. Symmetric inverse semigroup R_∞ is the semigroup of all infinite 0-1 matrices [g_ij] with at most one 1 in each row and each column such that g_ii=1 on the complement of a finite set. The binary operation in R_∞ is the ordinary matrix multiplication. It is clear that infinite symmetric group \mathfrakS_∞ is a subgroup of R_∞. The map ⋆:[ g_ij]↦[ g_ji] is an involution on R_∞. We call a function f on R_∞ positive definite if for all r₁, r₂, …, r_n∈ R_∞ the matrix [ f( r_ir_j^⋆)] is Hermitian and positive semi-definite. A function f said to be indecomposable if the corresponding ∗-representation π_f is a factor-representation. A class of the \mathfrakS_∞-invariant functions is defined by the condition f(rs)=f(sr) for all r∈ R_∞ and s∈\mathfrakS_∞. In this paper we classify all semifinite factor-representations of R_∞ that correspond to the \mathfrakS_∞-invariant positive definite functions.

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